Find the Sum of the Series 1+1/5+1/25+1/125

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by ${\displaystyle \frac{1}{5}}$ gives the next term. In other words, ${\displaystyle a_n=a_1{r}^{n-1}}$.

This is the form of a geometric sequence.

Substitute in the values of ${\displaystyle a_1=1}$ and ${\displaystyle r=\frac{1}{5}}$.

Multiply ${\displaystyle {\left(\frac{1}{5}\right)}^{n-1}}$ by ${\displaystyle 1}$.

Apply the product rule to ${\displaystyle \frac{1}{5}}$.

One to any power is one.

This is the formula to find the sum of the first ${\displaystyle n}$ terms of the geometric sequence. To evaluate it, find the values of ${\displaystyle r}$ and ${\displaystyle a_1}$.

Replace the variables with the known values to find ${\displaystyle S_4}$.

Multiply ${\displaystyle \frac{{\left(\frac{1}{5}\right)}^4-1}{\frac{1}{5}-1}}$ by ${\displaystyle 1}$.

Apply the distributive property.

Multiply the numerator by the reciprocal of the denominator.

Multiply ${\displaystyle \frac{-156}{125}}$ by ${\displaystyle \frac{1}{-1}}$.

Multiply ${\displaystyle 125}$ by ${\displaystyle -1}$.